Kawahara Lab.
Identification of Blasting Force Applied
to Tunnel Excavation
Hiroshi KOMINE∗ , Kazuya NOJIMA∗∗ and Naoto KOIZUMI∗∗∗
∗Department of Civil Engineering, Chuo University,Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, JAPAN
E-mail : h-komine− [email protected]
∗∗Department of Civil Engineering, Chuo University,Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, JAPAN
E-mail : [email protected]
∗ ∗ ∗Tunnel Group of Tech Sector, Sato Kogyo Co. Ltd, Hontyo 4-12-20, Chuo-ku, Tokyo 103-8630, JAPAN
E-mail : [email protected]
Abstract
This paper presents a parameter identification using the finite element method. In this research, the
elastic modulus is identified so as to minimize the performance function. Moreover, the external force
is also identified using the technique of the first order adjoint method, which is given by the square
sum of the difference between computed and observed velocities. The first order adjoint method is
one of technique of the inverse analysis. To calculate the velocity in the ground, the balance of stress
equation, the strain - displacement equation, and the stress - strain equation are employed. The ground
is assumed as elastic body with viscous damping. To identify the elastic modulus of the ground , the first
order adjoint method which is calculated from the velocity is used. In case to identify the parameter,
the gradient of the performance function is calculated using the adjoint equation. The Sakawa - Shindo
method is employed for the minimization algorithm. To solve the state equations, the Galerkin method
and the Newmark β method are employed for the state equation and the adjoint equation as spatial
and temporal discretizations.
Key words : Finite Element Method, First Order Adjoint Method,
Sakawa - Shindo Method, Parameter Identification
1
Introduction
and the value calculated using the finite element
method is assumed to be an analytical value, and
the sum of squares of the difference between the observation value and analytical value is calculated as a
performance function. It is shown that an analytical
value and the observation value become equivalent
when the performance function becomes 0. Because
an observation value and an analytical value depend
on the value of properties of the bedrock, the value of
physical properties of the bedrock can be requested
by using it as a value to perform the ground displacement velocity.
If this method could be applied to the practical
phenomenon, the safety and operating efficiency of
the construction will become drastically better. The
data of the ground displacement velocity and the
data of a lot of parameters that form the bedrock is
gathered on an actual construction site where digs
up the tunnel. And the practicality of this research
is verified. It will be able to be said that the effectiveness of this research is able to be proved though
a few improvements are still seen as a result.
In this research, the technology that forecasts the
ground properties and the stratum boundary (Young’s
modulus of geological condition) of the bedrock by
the three-dimensional numerical analysis technique
is developed by using the observation value of the
blast vibration wave digging up the bedrock.
In case that construction is designed, the ground
properties of the bedrock are investigated by the geologic reconnaissance of a drilling survey, a physical inquiry (elastic wave inquiry and electric inquiry,
etc.), and the paling of investigation and the rock
test for engineering properties, etc. However, if the
geologic structure is complex, it is a current state
that it is difficult to understand the properties of
the geological condition accurately. This technology
has aimed the prior forecast of the ground properties of the bedrock and improving the efficiency of
construction when under construction.
As the numerical analysis technique, the parameter identification method using the finite element
method is applied. The value in which the ground
displacement velocity generated by the blast vibration wave when digging up the bedrock in the construction site is assumed as the observation value
1
2
Basic Equation
Damping matrix and other matrices can be written as,
In this paper, indicial notation and summation
convention with repeated indices are used. The balance of stress equation is expressed as,
σij,j − ρbi + ρüi = 0,
Ciαkβ
Mαβ
(1)
whereσij , bi , ρ, üi denote total stress, body force,density
of the ground and acceleration, respectively. The
strain - displacement equation can be described in
the following form,
1
εij = (ui,j + uj,i ),
2
(2)
where εij and ui are strain and displacement, respectively. The stress - strain equation is
σij = Dijkl εkl ,
Γiα
V
(4)
(n+1)
ui
in which δij is Kronecker delta, and Lame’s constant
λ and µ are
νE
,
(1 − 2ν)(1 + ν)
Newmark β Method
In this paper, newmark β method is applied to
the finite element equation. In newmark β method,
velocity and displacement (n + 1) time cycle are expressed as follows;
where Dijkl express coefficient of elastic stress - strain
relation and can be written as,
λ=
(n+1)
u̇i
(5)
(n+1)
E
µ=
,
2(1 + ν)
where E and ν are the elastic modulus and Poisson
ratio, respectively. The boundary S can be divided
into SU and ST . On these boundaries, the following
conditions are specified.
= ûi
on SU ,
(7)
ti
= σij nj = t̂i
on ST ,
(8)
5
Finite Element Equation
Applying the finite element method, the discretization with the linear tetrahedral element is obtained
as follows;
Mαβ üiβ + Kiαkβ ukβ = Γiα .
(9)
(n)
üi ),
(16)
(n+1)
Performance Function
u̇i and u̇∗i are the computed and the observed velocity, respectively.
In the inverse analysis, calculation of the gradient of the performance function is necessary. The
extended performance function is composed of the
Considering the effect of damping, eq.(9) can be
expressed as follows;
Mαβ üiβ + Ciαkβ u̇kβ + Kiαkβ ukβ = Γiα .
=
(n)
+ üi ∆t
(n+1)
+γ∆t(üi
+
(15)
(n)
u̇i
To solve the inverse problem, the performance function is introduced. The performance function consists of quadratic sum of the difference between computed and observed state values. The purpose of this
research is to identify blasting force and the parameter of stratum, thus the first order adjoint method is
used as the inverse analysis. The method is defined
as finding the optimal value so as to minimize the
performance function. The performance function is
expressed as follows;
Z
1
(u̇i − u̇∗i )Wij (u̇j − u̇∗j )dt,
(17)
J=
2 t
where the ûi and t̂i mean the known values on the
boundary and ni is the external unit vector to the
boundary. The blasting force can be expressed by ti
applied to the tunnel face, which is included on ST .
3
1 (n)
(n)
+ u̇i ∆t + üi ∆t2
2
(n+1)
(n)
+β∆t2 (üi
+ üi ),
(n)
= ui
where ui
, u̇i
are substituted for the finite element equation. Acceleration at (n + 1) cycle is sub(n+1)
stituted into eqs. (15) and (16) to calculate ui
,
(n+1)
u̇i
. In this research, β and γ are assumed as 0.25
and 0.50, respectively.
(6)
ui
ST
in which Nαi is the linear interpolation function for
the finite element method.
4
(3)
Dijkl = λδij δkl + µ(δik δjl + δil δjk ),
Kiαkβ
= α0 Mαβ + α1 Kiαkβ ,
(11)
Z
(Nα ρNβ )dV,
(12)
=
ZV
=
(Nα,j Dijkl Nβ,l )dV,
(13)
ZV
Z
=
(Nα ρbi )dV −
(Nα ti )dS, (14)
(10)
2
of the extended performance function J ∗ instead of
the performance function J. The first variation of
the extended performance function J ∗ is expressed
as follows;
Z
∗
δJ
=
(u̇i − u̇∗i )Wij δuj dt
t
Z
+
δλTiα (Fiα − Mαβ üiβ
performance function and the basic equation by being multiplied by the Lagrange multiplier. Taking
the first variation of the extended performance function, the gradient of the performance function can
be calculated. Using the gradient, the objective parameter is updated by the iterative calculation.
6
Weighted Gradient Method
t
− Ciαkβ u̇kβ − Kiαkβ ukβ )dt
Z
λTiα (δFiα − Mαβ δüiβ
+
The weighted gradient method is applied to the
minimization technique. the modified performance
function is expressed as follows;
Z
∗
K = J + (X n+1 − X n )W (X n+1 − X n )dt. (18)
t
− Ciαkβ δ u̇kβ − Kiαkβ δukβ )dt.
Considering each term of the first variation of the
extended performance function, δJ ∗ equals to 0. Then,
the adjoint equation can be derived as follows;
t
Differentiating the both sides of eq.(18) with respect to X n , the optimal condition of the modified
performance function can be expressed as
∂K
= 0.
∂X
δJ ∗
= (u̇(tf ) − u̇∗ (tf ))Wij − (u̇(t0 ) − u̇∗ (t0 ))Wij
(19)
−λTiα (tf )Mαβ δ u̇iβ (tf ) + λTiα (t0 )Mαβ δ u̇iβ (t0 )
The gradient of the performance function is calculated by inverse analysis. X and W are the parameter to be identified and the weighting function,
respectively.
X n+1 = X n − grad(J ∗ )/W,
+λ̇Tiα (tf )Mαβ δuiβ (tf ) + λ̇Tiα (t0 )Mαβ δuiβ (t0 )
−λTiα (tf )Ciαkβ δukβ (tf ) + λTiα (t0 )Ciαkβ δukβ (t0 )
Z
Z
− λ̇Tiα Mαβ δuiβ dt + λ̇Tiα Ciαkβ δukβ dt
t
Z
Zt
T
− λiα Kiαkβ δukβ dt + δλTiα (Fiα − Mαβ üiβ
t
t
Z
−Ciαkβ u̇kβ − Kiαkβ ukβ )dt − (üi − ü∗i )Wij δudt
t
Z
T
+ λiα δFiα dt
(20)
Using eq.(20), the parameter is updated by the iterative cycle.
7
First Order Adjoint Method
t
= 0.
In this research, to identify the parameters, like as
the impulse wave broken out by a blasting excavation
and elastic modula of strata, the first order adjoint
method is efficiently employed.
The extended performance function is expressed
as follows;
Z
1
(u̇i − u̇∗i )Wij (u̇j − u̇∗j )dt
J∗ =
2 t
Z
+
λTiα (Fiα − Mαβ üiβ
(23)
Eq.(23) is the first variation of the extended performance function. The first order adjoint equation
and the terminal condition of the Lagrange multiplier can be extracted from eq.(23). The first order
adjoint equation can be expressed as follows;
Mαβ λ̈iβ − Ciαkβ λ̇kβ + Kiαkβ λkβ
+(üi − ü∗i )Wij = 0
on SU .
t
− Ciαkβ u̇kβ − Kiαkβ ukβ )dt.
(21)
(24)
To calculate the adjoint equation, It is needed to
obtain the terminal condition. The terminal time
is denoted by tf . The terminal conditions are also
able to calculate t0 solving the equation that the
first variation of the extended performance function
equal to 0. The terminal conditions at the time of
tf are shown as follows;
In this research, the objective parameters are assumed as the external force t or elastic modulus E.
The first order adjoint method of parameter identification is introduced in following two subsections.
7.1
(22)
Discretization
(External Force)
The minimization condition with constraint condition results in satisfying the stationary condition
3
λi (tf ) = 0,
(25)
Mαβ λ̇iα (tf ) + (u̇i (tf ) − u̇∗i (tf ))Wij = 0.
(26)
Eqs.(25) and (26) are the terminal condition of
the adjoint equation. On the other hand, λ̈i (tf ) is
the terminal condition of the acceleration of the Lagrange parameter must be calculated as using λ̇i (tf )
and λi (tf ). Using the first adjoint equation, the
gradient of the extended performance function can
be calculated. The gradient of the extended performance function, grad(J ∗ ) is expressed as follows;
350000
Pressure[kPa]
300000
Pressure(kPa)
250000
200000
150000
100000
50000
0
grad(J ∗ ) = λi
0
(27)
= (u̇(tf ) − u̇∗ (tf ))Wij − (u̇(t0 ) − u̇∗ (t0 ))Wij
−λTiα (tf )Mαβ δ u̇iβ (tf ) + λTiα (t0 )Mαβ δ u̇iβ (t0 )
+λ̇Tiα (tf )Mαβ δuiβ (tf ) + λ̇Tiα (t0 )Mαβ δuiβ (t0 )
−λTiα (tf )Ciαkβ δukβ (tf ) + λTiα (t0 )Ciαkβ δukβ (t0 )
Z
Z
− λ̇Tiα Mαβ δuiβ dt + λ̇Tiα Ciαkβ δukβ dt
t
Z
Zt
T
− λiα Kiαkβ δukβ dt + δλTiα (Fiα − Mαβ üiβ
t
t
Z
−Ciαkβ u̇kβ − Kiαkβ ukβ )dt − (üi − ü∗i )Wij δudt
t
Z
T
−βt
−αt
+ λiα (e
− e )δAdt
(28)
t
= 0.
In(β/α)
.
β−α
(31)
The first order adjoint equation and the terminal
condition are
Mαβ λ̈iβ − Ciαkβ λ̇kβ + Kiαkβ λkβ
(29)
+(üi − ü∗i )Wij = 0
on SU .
where A, α and β are constants. A controls the
peak value of F (t), α and β control the shape of
F (t), where α and β are assumed as 1000 and 5000,
respectively.
Thus, in this numerical study, the
parameter which must be identified is A. When the
parameter A is assumed as an objective parameter,
the shape of explosion is approximated to the practical one. α, β can control the peak time of external
force. The peak time can be control to calculate the
follow equation.
tpeak =
0.005
δJ ∗
where F0 , Vd and ρe are peak detonation pressure,
detonation velocity of the explosive and specific gravity of the explosive, respectively.
Usually the dynamic borehole pressure rises very
rapidly, achieving its peak within 1 millisecond and
then decays exponentially with time as schematically
shown in Fig.1. A combination of two exponential
functions is assumed to represent the time variation
of dynamic borehole pressure can be written as follows;
F (t) = A(e−αt − e−βt )
0.004
The gradient of extended performance function is
transformed because the parameter has been changed.
As eq.(29) is applied to eq.(23), the first variation of
the extended performance function can be written
as follows;
The peak detonation pressure of a specific explosive, the peak dynamic pressure action at the cavity
surface, is given by the U.S. Army Corps of Engineers (1972) as
ρ e Vd 2
)
1 + 0.8ρe
0.003
Fig.1 Time History of Pressure
Pressure of Explosives
F0 = 6.06 × 10−3 (
0.002
Time(s)
grad(J ∗ ) is the gradient of the extended performance function. The purpose of this study is to
identify the external force created by the dynamite
shock at the blasting excavation. To identify the parameter of Fi , the gradient of extended performance
function is used in the minimization technique.
7.2
0.001
(32)
λi (tf ) = 0,
(33)
Mαβ λ̇iβ (tf ) + (u̇i (tf ) − u̇∗i (tf ))Wij = 0.
(34)
The gradient of extended performance function is
expressed as follows;
grad(J ∗ ) = λi (e−βt − e−αt ).
(30)
Fig.1 shows the time history of pressure using eq.(29)
4
(35)
As using eqs.(31),(32),(33) and (34) in the backward analysis, the parameter A can be identified.
If this method can be applied to the practical phenomenon, this technology can be proved as effective
one.
8
8.1
Ikawa Tunnel
Mesh Generation
The finite element mesh is made by Dr.Kazuya
NOJIMA.
The data of each strata are observed at Ikawa tunnel (Tokushima-ken). To apply this technology to a
practical phenomenon, a mesh which is generated
using the practical data is needed. Fig.2 shows the
topography of Ikawa tunnel.
Fig.4 Counter of Ikawa Mesh
The finite element mesh is made so as to reduce the
number of nodes and elements. Fig.5 is used for the
analysis mesh.
Fig.2 Topography of Ikawa Tunnel
As using this figure, analytic domain is decided.
Fig.3 shows the analytic domain.
Fig.5 Finite Element Mesh
8.2
Parameter Identification
In the numerical study, parameter identification of
the external force A is carried out. The initial values
of the external force Fi are set as 1.0 × 106 [kN/m2 ].
And the target value is assumed as 1.0×105[kN/m2 ].
The Poisson ratio , elastic modulus, density of the
ground, and dt are 0.32, 1.6767 × 106[kN/m2 ], 2.3 ×
103 [kg/m3 ] and 1.0 × 10−3 [s], respectively.
12000
Performance Function
11000
10000
Performance Function
Fig.3 Analytic Domain
As using Fig.3, the finite element mesh can be generated. Figs.4 and Fig.5 show the counter of the
computational domain and the finite element mesh,
respectively.
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0
20
40
60
80
100
120
140
Iteration
Fig.6 History of Performance Function
5
there are several maximums of the amplitude seen
after 0.1 seconds, or there is nothing. The computed
and observed velocities are well in agreement. It is
thought that it led to obtain a well agreement of
giving the external force as shown in Fig.1.
1.1e+06
A
1e+06
900000
800000
700000
A
600000
500000
400000
300000
9
200000
100000
Conclusion
0
0
20
40
60
80
100
120
140
The computational value is well agreement with
the observed velocity. The reason why is the external force is assumed to be given like as Fig.1. It
can be proven that the shape of the observed velocity can be computed. Therefore, the parameter
identification of the external force is proven to be
possible. The final purpose of this research is to
identify the elastic modulus of each layer. In this research, there are several unknown parameters like as
the elastic modulus, the external force, the damping
coefficients, and so on. The external force can be
proven to be known parameter. Therefore, the elastic modulus is the next parameter to be identifed.
The computational results that the elastic modulus
of each layer is to be identifed is carried out.
Iteration
Fig.7 History of External Force (A)
Figs.6 and .7 show the history of performance function and that of external force, respectively. The external force can be converged to 0 from initial value
to target value. The observed value used in this
study is computed like as computed value. Fig.8
show the prediction model of the finite element mesh
predected by the value of the elastic moduli.
References
[1] A. Hikawa, M. Kawahara, and N. Kaneko : Parameter Identification of Ground Elastic Modulus at Excavation Site of Tunnel, Vol.1, Research
Report of Professor M. Kawahara Lab., pp7283(2004)
Fig.8 Prediction Model
[2] Guoxi Wu : Dynamic Response Analysis of Saturated Granular Soils To Blast Loads using a Single Phase Model, A Research Report Submitted
to NSERC, (1995)
The computed velocity is calculated and is compared with the value of the observed one. Fig.9 show
the history of velocity of computed and observed velocities.
30
Observed Velocity
Computational Velocity
25
20
Velocity [m/s]
15
10
5
0
-5
-10
-15
-20
-25
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time [s]
Fig.10 Comparison between the
Computed and Observed Velocities
In this research, the external force is given in one
time. However, the practical external force is given
five times. The difference among both is whether
6